Importance of High Angular-Momentum Channels in Pseudopotentials for Quantum
Monte Carlo
William W. Tipton,1 Neil D. Drummond,2, 3 and Richard G. Hennig1
1
arXiv:1203.5458v1 [cond-mat.mtrl-sci] 25 Mar 2012
Department of Materials Science and Engineering,
Cornell University, Ithaca, New York 14853, U.S.A.
2
Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
3
TCM Group, Cavendish Laboratory, University of Cambridge,
J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
(Dated: December 30, 2013)
Quantum Monte Carlo methods provide in principle an accurate treatment of the many-body
problem of the ground and excited states of condensed systems. In practice, however, uncontrolled
errors such as those arising from the fixed-node and pseudopotential approximations often limit
the quality of results. We show that the accuracy of quantum Monte Carlo calculations is limited
by using available pseudopotentials. In particular, it is necessary to include angular momentum
channels in the pseudopotential for excited angular momentum states and to choose the local channel
appropriately to obtain accurate results. Variational and diffusion Monte Carlo calculations for Zn,
O, and Si atoms and ions demonstrate that these issues can affect total energies by up to several eV
for common pseudopotentials. Adding higher-angular momentum channels into the pseudopotential
description reduces such errors drastically without a significant increase in computational cost.
I.
INTRODUCTION
Computational electronic structure methods have been
extremely useful in developing our understanding of the
atomic and electronic structures of real materials. As
methods have become more accurate and their implementations increasingly efficient, simulation and calculation
have taken some of the burden of finding and characterizing new materials off experimental work.1
Density functional theory (DFT), in particular, has
been widely applied to many systems in recent decades.
It is computationally efficient compared to other methods
with similar accuracy, and robust, user-friendly software
packages have made the method easy to apply. However, its accuracy is still insufficient for some applications, and the lack of a systematic way to improve its
results or estimate its errors has hampered progress. In
particular, electron correlation effects can be significant
in many complex materials and are not captured accurately by many of the commonly-used density functionals. The development of functionals which accurately
describe the electronic band gap, van der Waals interactions, and other electronic properties of materials is still
an active area of research.2–5
These issues are overcome by methods which treat
quantum many-body effects explicitly from the outset
such as quantum Monte Carlo (QMC). QMC methods
are among the most accurate many-body methods and
can reliably and accurately predict ground-state expectation values for many systems and have often been used
as a benchmark for DFT work.6 Among the quantum
Monte Carlo methods, variational Monte Carlo (VMC),
diffusion Monte Carlo (DMC), and auxillary-field quantum Monte Carlo7 are the most mature in terms of applicability to solid state systems. We treat only VMC
and DMC in this work and refer to them collectively as
QMC.
As computers become faster and high-quality software packages for QMC such as CASINO8 , QMCPACK9 ,
QWALK10 , and CHAMP11 mature, these calculations
are becoming less challenging. It is therefore important
to identify and propagate the best-practice procedures
for performing these calculations as they become more
routine.
QMC and other correlated-electron methods, particularly for heavy elements, usually employ the pseudopotential approximation to reduce the computational cost.
The common form is the non-local, norm-conserving
pseudopotential12 which includes different potentials for
each angular-momentum component of the wavefunction.
In this work, we determine the error in the energy due
to an insufficient number of angular-momentum channels
in the pseudopotential and discuss other sources of error
in QMC calculations. We show that pseudopotentials
which include channels to account for higher angularmomentum components of the wavefunction are necessary for performing accurate pseudopotential calculations
in QMC. Such pseudopotentials are not the norm in the
literature, and we suggest that this be corrected in order
that QMC methods be suitable for routine applications
to scientifically and technologically interesting systems.
II.
BACKGROUND
The computational cost of all-electron QMC scales approximately as Z 5.5 or Z 6.5 with respect to the atomic
number.13,14 This scaling makes the direct application of
all-electron QMC to heavy atoms difficult. In practice,
many properties of atoms are primarily due to the behavior of and interactions between valence electrons, and
so a pseudopotential approximation is commonly used to
remove core electrons from the calculation and reduce
2
the necessary computational effort.
Modern pseudopotentials are non-local in the sense
that they act differently on distinct angular-momentum
components of the wavefunction. This is necessary to accurately capture the effects of the nucleus and core electrons on the valence electrons since the pseudopotential
not only represents the effective electrostatic potential
but also enforces orthogonality of the valence orbitals to
the lower-energy states of the same angular momentum.
Of course, there is no clear distinction between core
and valence electrons in many-body methods as the electrons are indistinguishable particles. Thus, the application of pseudopotentials does neglect exchange and correlation interactions between the valence and removed
core electrons as well as that between the core electrons
themselves. These errors are not explicitly accounted for
in the calculations. However, the core-core interactions
largely cancel out when considering energy differences,
and the core-valence interactions may be kept small by
a choice of core size which leads to significant spatial
separation between the core and valence electron densities. These techniques can lead to results as accurate as
all-electron calculations.14 Additionally, the use of corepolarization potentials can account for some core-valence
correlations.15–17
To introduce non-local pseudopotentials in QMC, the
electron-ion potential of any given atom is divided into a
local potential V̂loc which is applied to the whole wavefunction and several corrections V̂nl which account for
the difference between the local potential and those which
should be seen by the individual angular-momentum
components of a wavefunction:
V̂loc + V̂nl =
Nel
Nel
X
X
ps
ps
V̂ .
V (ri ) +
loc
nl,i
i=1
(1)
i=1
ps
acts on a function f (ri ) by
nl,i
Z
X ps
ps
∗
V (ri )Ylm (Ωi )
Ylm
(Ω0i )f (r0i )dΩ0i ,
V̂ f (ri ) =
nl,i
nl,l
4π
The nonlocal operator V̂
l,m
(2)
where the angular integral in the operator projects the
wavefunction onto spherical harmonics. Each angularmomentum component thus “feels” its own potential,
ps
V (r), which is only a function of the electron-nuclear
nl,l
distance r and accounts for the difference between the
ps
desired l-dependent potential and the local channel V .
loc
ps
The local potential, or local channel, V (r), is by conloc
vention chosen to be the exact potential applied to one
of the angular-momentum components, and so sum in
Eq. (2) need not include this local component.6
The choice of local channel itself is arbitrary but is
often chosen for convenience during the pseudopotential design process. In particular, judicous choice of the
local channel is often necessary to avoid the problem
of ghost states which can arise due to the KleinmanBylander transformation.18 We will see that the same
choice of local channel that is suitable for that transformation may not be the best with regards to accuracy of
QMC calculations.19
In an independent electron theory such as HartreeFock or DFT, atomic wavefunctions are composed of
some number of the lowest-energy single-particle orbitals. For example, in these frameworks, the electronic configuration of an oxygen atom may be written as 1s2 2s2 2p4 . Notice that this wavefunction contains no angular-momentum components above l = 1.
Thus, a non-local pseudopotential in the above form
which acts on these single-atom wavefunctions need not
ps
include terms V
for l ≥ 2 if it is to be used to calculate
nl,l
ground-state atomic properties.
The situation is not so simple in the case of solids
and other extended systems where changes in the wavefunctions due to bonding effectively introduce higher
angular-momentum components. Indeed, in the case of
systems such as Si and other second row elements, one
may find wavefunctions with significant higher angularmomentum character. In this case, it may be necessary
to use a pseudopotential with a d-channel when studying these systems in DFT, especially in the high-pressure
regime. However, these errors often cancel when considering energy differences and are frequently neglected in
practice.20,21
In QMC and other correlated-electron methods, excitations of the wavefunction into higher angular-momentum
states arise immediately, i.e., even for atoms. In VMC,
wavefunctions may be represented by the product of a
Slater determinant of single particle orbitals and the socalled Jastrow factor. The Jastrow is a positive function
of inter-particle distances, and its purpose is to directly
account for many-body correlation effects. Naturally, the
VMC wavefunction is then no longer entirely composed
of the lowest-order spherical harmonics. The situation
in DMC (as well as other correlated-electron methods) is
analogous.6
Notice from Eq. (1) that, in the absence of a pseudopotential channel to deal with the higher-angular momentum components of the wavefunction, these components simply feel the local channel. This is incorrect and
may be drastically so, especially in the case where the
local channel was designed to enforce orthogonality to
the lower-energy orbitals with a particular angular momentum. This can lead to sizeable errors in total energy
calculations.
Now, this effect is not a particularly surprising one and
certainly has been understood by some in the density
functional theory community since the beginnings of the
use of pseudopotentials in that field (see, for example,
Ref. 22). However, inclusion of so-called higher angularmomentum channels is not the normal practice in the
development of potentials for use with QMC.
There are a limited number of pseudopotentials
available for use with QMC. In particular, the application of projector-augmented waves23 or ultra-soft
pseudopotential24 techniques in QMC is currently not
3
TABLE I. Choices of angular-momentum channels and local
channels for the various pseudopotentials considered for oxygen, silicon and zinc.
O
Si
Zn
Standard
Channels
Local
s, p
s or p
s, p
s or p
s, p, d
s or d
Augmented
Channels
Local
s, p, d
s or p
s, p, d
s or p
s, p, d, f
s, d, or f
feasible since the DMC operators for the projectors and
the augmentation charge are unknown. However a number of semi-local pseudopotentials have been developed
with QMC applications in mind. Greeff et al. developed a carbon pseudopotential which included s- and pchannels.25 Ovcharenko et al. applied a similar methodology to produce pseudopotentials for Be to Ne and Al
to Ar with lmax = 1.26 Burkatzki et al. present potentials for many of the main group elements27 and for the
3d transition metals.28 Their Si and Zn potentials have
3 channels, and their O potential has 2. These authors
all cite the rule of thumb that lmax should be at least
as high as the highest angular-momentum component in
atomic core. Trail et al. developed a variety of pseudopotentials for all elements from H to Hg. These all have
exactly 3 channels and are associated with the CASINO
code which, until recently, only supported pseudopotentials with exactly 3 channels.29
III.
METHODOLOGY
We determine how the number of channels and the
choice of local channel affects the energy for several atoms
and ions. We compute the total energies and first and
second ionization energies of the zinc, oxygen, and silicon
atoms using several related pseudopotentials. These elements provide interesting test cases due to their varied
electronic structures. Additionally, we are interested in
the application of QMC methods to bulk semiconductors
such as Si and ZnO.30,31
The oxygen and silicon pseudopotentials are based on
those by Driver et al.32 , and the zinc pseudopotential is
based on that by Bennett and Rappe.33 All three are
generated using the Opium pseudopotential code.34 Cutoff radii and basis functions for the construction of the
pseudopotentials were chosen to minimize the difference
in pseudopotential and all-electron valence energy levels
calculated in DFT using the PBE exchange-correlation
functional35 for several electronic configurations.
Table I lists the angular-momentum channels and the
choice of local channel for each of our pseudopotentials.
For each element, we consider (i) pseudopotentials with
the minimum number of channels (s and p for Si and O;
s, p and d for Zn) and (ii) pseudopotentials that contain an additional angular-momentum channel (d for O
and Si; f for Zn). We refer to the first set as standard
FIG. 1. (color online) Pseudopotentials for O, Si, and Zn.
pseudopotentials and the second one as augmented pseudopotentials. For the local channel we consider the s
and p channels for Si and O and the s, d or f channels
for Zn. This results in a total of 13 pseudopotentials
as listed in Table I. Figure 1 shows the distance dependence of the angular momentum channels for the various
pseudopotentials. For Si and Zn, we confirmed that the
pseudopotentials accurately describe the lattice parameters of the ground state crystal structure and for O, we
confirmed that the pseudopotential reproduces the dimer
bond length at the DFT level.
QMC calculations were performed using the CASINO
code.8 We implemented support for pseudopotentials
with an arbitrary number of angular-momentum channels in CASINO. Total energy calculations are performed on the 9 isolated ions with each of the applicable pseudopotentials. The VMC calculations used
Slater-Jastrow variational wavefunctions with orbitals
expressed in a blip basis.36 The single-particle orbitals
were generated using the PWSCF code37 and the PBE
exchange-correlation functional.35 Plane-wave cutoffs of
70 Ry for oxygen and silicon and 100 Ry for zinc
were used to converge the total energies to 2 meV.
The known magnetic states of the atoms and ions were
used. The Jastrow factor is a non-negative function of
inter-particle distances and includes two-body electronelectron and electron-nucleus and three-body electronelectron-nucleus terms as implemented in CASINO.38
The backflow transformation39 was not found to pro-
4
vide any significant benefit in these cases. Parameters
were added to the Jastrow factor of the trial wavefunction gradually during its optimization. The Jastrow parameters were optimized using variance minimization40
followed by energy minimization in the final step.41 Trial
wavefunctions were evaluated by their mean energy plus
three times the statistical error in the energy, following
Ref. 42.
Several additional details of our VMC calculations are
noteworthy. First, the integral in Equation (2) is performed on a spherical grid in real space. This integration mesh must be chosen to be sufficiently dense to accurately calculate the contributions to the energy from
higher angular-momentum compontents of the wavefunction and thus evaluating the effects which are the focus
of this paper. Secondly, it is the default behavior of the
CASINO code that the non-local contributions to the
energy are assumed constant and are not re-calculated
during a variance minimization step. In many systems,
this improves the runtime of the algorithm significantly
while still giving good results — in some cases it actually
improves the performance of the variance minimization.
However, as we will see, the non-local contributions are
significant in many of our calculations. We found it necessary in many cases to re-calculate the non-local contributions to the energy at each step of the optimization
to ensure the stability of the optimization process during
Jastrow optimization.
Our DMC calculations were performed using the pseudopotential locality approximation.43 For each system,
we performed 1, 000 equilibration and 3, 000 accumulation steps on each of 256 processors with a timestep of
0.01 Ha−1 and a target population of 2, 000 walkers.
Finally, atomic ionization energies are simply differences between the total energies of the appropriate
species.
IV.
FIG. 2. (color online) Total VMC energy in Hartree and
statistical error in the energy of each species with respect to
each Hamiltonian. Pseudopotentials are denoted according to
the choice of local channel and as ‘aug’ if they are augmented
with an additional channel or ‘std’ otherwise.
RESULTS AND DISCUSSION
Figures 2 and 3 show the energies for each ionpseudopotential combination for VMC and DMC, respectively. The error bars indicate only the statistical uncertainties in the energies associated with the QMC calculations. First, it is important to notice the axes. The
variation in the total energies differs between the species.
For Si, it is on the order of milli-Hartrees, while for Zn, it
is on the order of tenths of Hartrees. O falls somewhere
in between.
Consider now the VMC and DMC total energies. The
two sets of data exhibit similar trends which is to be expected since they arise from the differences in the pseudopotentials. The first thing to notice is that the calculations using an augmented pseudopotential (i.e., the two
or three left-most data points in each panel) are largely in
agreement with each other, while this is not the case for
the standard potentials. That is, the choice of local channel has a large effect on the total energy when using the
FIG. 3. (color online) Total DMC energy in Hartree and
statistical error in the energy of each species with respect to
each Hamiltonian. Pseudopotentials are labeled as in Fig. 2.
standard pseudopotentials since the higher-l components
of the wavefunction see that local channel. When using
the augmented potentials, more of the wavefunction sees
its correct potential, and the choice of local channel has
less effect on the result of the calculation.
Indeed, if we take the calculations with augmented potentials to indicate the correct result, we can understand
the errors in the other total energies in terms of which
potentials are incorrectly applied to certain components
of the wavefunction. As seen in Figure 1, the s-channel is
the most repulsive for each of the species, the p-channel
is in the middle, and the d-channel is the most attractive.
The f -channel is slightly above d-channel in the case of
5
TABLE II. Lowest-energy excitations in eV to higher-l states
for each species from experiment.44–47
Species
Neutral
Singly-Ionized
Doubly-Ionized
FIG. 4. Variance in local energy in atomic units and associated statistical error of each VMC calculation. Hamiltonians
are labeled as in Figure 2.
FIG. 5. Comparison of the ionization energies in eV for oxygen, silicon and zinc in DFT, VMC and DMC for the different choices of pseudopotential with experiment. Energies for
oxygen and silicon include corrections for the pseudopotential
error at the DFT level as described in the text.
zinc.
Thus, we expect that calculations in which components
of the wavefunction incorrectly see the s-channel to be
too high in energy. Indeed these data points (which are
the second right-most point in each frame of the total energy plots) exhibit this trend. Similarly, the right-most
data point in each frame corresponds to a calculation
wherein any higher l component of the wavefunction sees
the d-channel, and these results are erroneously low in
energy. Even the residual differences between the energies calculated using the augmented potentials follow this
trend. This is indicative of small amounts of yet-higher
O
12.08
28.7
40.23
Si
5.86
9.84
17.72
Zn
8.53
14.54
31.9
l character in the wavefunctions.
We can get some idea of the amount of virtual excitations that might be present in the many-body ground
state of each of the species by considering the energies
of these excitations. In this way, we can understand the
expected magnitude of these effects.
Table II presents the lowest energy excitation to a
higher angular-momentum state for each of the three
atoms for the various charge states. The excitation energies of these states increase with the level of ionization. Additionally, the d levels are relatively high in
oxygen but low in silicon. The f levels in Zn trend in
between. Thus, we expect that the effects in total energy
described in this paper will be especially pronounced for
the neutral species relative to the positive ones and for
silicon relative to oxygen. Note that this effect due to the
lower excitation energies here has implications not only
for the atomic wavefunctions. Lower-energy states are
more likely to participate in bonding in molecules and
solids, and it is especially important to design pseudopotentials to account for that.
Indeed, for the silicon species, the decreasing significance of the extra channel with increasing charge is clear.
This effect is less readily apparent in oxygen and zinc
data, and is likely obscured by another notable and correlated effect. The variance in the local energy for each
VMC calculation is shown in Figure 4. Eigenstates of
the Hamiltonian have a variance of zero, and higher variances indicate worse approximations of the ground-state
wavefunction. That is, higher variances are correlated
with higher total energies. From the figure, it is clear
that our calculations using the standard pseudopotentials
with the s-channel local resulted in poorer-quality wavefunctions. These higher variances are not fundamental
properties of the system but demonstrate that it is more
difficult to optimize a Slater-Jastrow wavefunction with
respect to these less realistic Hamiltonians. Additionally,
the wavefunctions might be better described by a multideterminant expansion, especially in the case of oxygen.
Finally, the first and second ionization energies for each
element are shown in Figure 5. The DFT ionization energies were computed from the same calculations used
to create the orbitals for the VMC trial wavefunction,
and finite-size effects due to periodic boundary conditions
were treated using the method of Makov and Payne.48
Several sources of errors may explain the deviation of
these results from experimental values. First, we calculated spin-orbit corrections to the total energies at the
DFT level and found that they largely cancel in the ion-
6
ization energies, resulting in corrections too small to account for the observed differences in the ionization energies between QMC, DFT and experiment.
Second, the pseudopotential approximation itself leads
to several errors other than those focused on in this paper. By removing explicit treatment of core electrons
from the calculation, we are neglecting correlations between the core and valence electrons. This is minimized
but not altogether eliminated by designing pseudopotentials so that the core and valence electrons are spatially
separated. The core-valence correlation may be particularly important for the case of zinc where the 3d valence
electron state have a sizeable spatial overlap with the 3p
core electron states and may explain the large errors in
the ionization energies. Thirdly, evaluation of the pseudopotentials in DMC is subject to the locality approximation43 used in this work or the lattice-regularized method
by Casula.49 Pozzo and Alfè50 found that, in magnesium
and magnesium hydride, the errors of the locality approximation and the lattice-regularized method are comparably small, but that the lattice method requires a much
smaller DMC time step.
Finally, our pseudopotentials themselves could likely
be further optimized within the same framework. In particular, the scattering properties of PBE pseudopotentials may be poor at certain energy scales, and HF potentials might perform better in conjunction with correlatedelectron methods.25 To test the accuracy of the pseudopotential approximation for oxygen, we performed an
all-electron, single-determinant DMC calculation of an
isolated oxygen atom with a Slater-Jastrow trial wavefunction and found an ionization energy of 13.611(7) eV,
in close agreement with the experimental value. This
strongly suggests that the errors in the DMC ionization
energies in Fig. 5 are due to the pseudopotentials.
With this in mind, we estimated corrections for the
pseudopotential error at the DFT level for oxygen and
silicon. All-electron DFT/PBE ionization energies were
calculated using the Gaussian code51 converged with respect to the atomic basis. The difference between these
ionization energies and those found with PBE using pseudopotentials should capture much of the error due to the
pseudopotentials, and we have added these differences to
the QMC results for oxygen and silicon shown in Figure
5.
In the case of zinc, the error in the ionization energy in
QMC stems from the poor description of the 3d-levels of
the zinc atom in DFT. Semilocal functionals are known
to place the 3d level of the Zn atom significantly too
high52,53 . This results in an incorrect description of the
d-channel of the pseudopotential and of the 3d-orbital in
the trial wave function which is reflected in both the large
energy variance and large deviation of the QMC ioniza-
1
W. Kohn, Nobel Lectures, Chemistry 1996-2000 (1999).
tion energy from experiment. Correcting for this error at
the DFT level by adding the energy difference between
an all-electron and pseudopotential DFT calculation to
the QMC results is found to be insufficent.
V.
CONCLUSIONS
We showed that pseudopotentials which include channels to account for higher angular-momentum components of the wavefunction are necessary for performing
accurate pseudopotential calculations in QMC. For O,
Si and Zn, we determined how the number of angularmomentum channels and the choice of local channel in
the pseudopotential affects the total energy and ionization energies of these atoms in QMC. We find a sizable
error in the total energies for any choice of local channel
when the pseudopotentials do not include at least one
additional angular-momentum channel above the highest angular-momentum component of the ground state
wavefunction of the atom. This is because, contrary to
single-electron mean-field methods such as DFT and HF,
atomic ground state wavefunctions in correlated-electron
methods include higher angular-momentum character.
These components effectively see the wrong potentials
when using standard pseudopotentials. This situation is
expected to be even more pronounced in the case of solids
and molecules.
Our results suggest that the best practice is to include at least one channel in the pseudopotential above
the highest angular-momentum component of the ground
state wavefunction in single-particle methods. Additionally, this highest channel should be used as the local
channel as it will generally be most similar to missing,
yet-higher angular-momentum channels.
VI.
ACKNOWLEDGMENTS
The authors would like to thank John Trail and
Richard Needs for helpful discussions. This research has
been supported by the Cornell Center for Materials Research NSF-IGERT: A Graduate Traineeship in Materials for a Sustainable Future (DGE-0903653), by the
National Science Foundation under Award Number CAREER DMR-1056587, and by the Energy Materials Center at Cornell (EMC2), funded by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences
under Award Number de-sc0001086. This research used
computational resources of the Texas Advanced Computing Center under Contract Number TG-DMR050028N
and of the Computation Center for Nanotechnology Innovation at Rensselaer Polytechnic Institute.
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